Differentiate $(5 x)^{3 cos 2 x}$ with respect to $x .$

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Solution

Let $y = (5 x)^{3 cos 2 x}$
$⟹ log y = 3 cos 2 x log (5 x)$
Differentiating with respect to $x$
$\frac{1}{y} \frac{d y}{d x} = - 6 sin 2 x log (5 x) + \frac{cos 2 x}{x}$
$\frac{d y}{d x} = (5 x)^{3 cos 2 x} (- 6 sin 2 x log (5 x) + \frac{cos 2 x}{x})$

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